QUESTION: In a formula to find the curvature of a special kind of a spherical mirror... what do the letters λ, D and b mean?
I believe Ra is radius of curvature.
The image attached contains the actual formula.
ANSWER: The typical equations of curvature of a spherical mirror are here: http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/mireq.html
I am not familiar with what "special" kind of spherical mirror you are referring to, nor am I familiar with this equation. What makes it special? What is the rest of the context of the equation? Lambda would typically be a wavelength, which should be immaterial for pure mirrors.
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QUESTION: It's an astigmatic mirror.
The equation is found here:
This is the only relationship I can find for the curvature of an astigmatic mirror online. I'm not sure if this can be used as a general equation for any astigmatic mirror.
ANSWER: Well, this is a very specific type of special. Astigmatism, a term you may have heard referring to vision, is caused by and uneven curvature of a focusing element in one axis vs another. The equation above does indeed involve the wavelength of light, and is very different than a spherical mirror. If you're asking me this level of question without reading the text (VERY COMPLEX) that you sent, I'm afraid that equation is out of your depth for now. See here for something on astigmatic diffraction, there are many more references you can google...but why do you want to know about such an obscure and complex topic? And why this equation about submillimeter waves? ftp://ftp.ccmr.cornell.edu/utility/FEI%20temp/Morgagni%203.0/Morgagni/doc/morgagnihelp/tem/stigmate.htm I still need more context to help with your real question, you couldn't possible need the one you sent...it makes no sense.
I do want to answer your question, but I don't want to type up anything that will confuse the issue without knowing more about the question and how you ever found this equation in the first place... it's very obscure! That means it's probably not relevant to what you actually want to know...which is the question I want to get at and answer.
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QUESTION: Well yes, I stumbled upon it while looking for an equation to find zoom ratios at different points of an astigmatic mirror. And I'm sure you're right; the above mentioned equation must not be what I'm looking for.
Maybe if I treat it as two normal spherical mirrors for the two axes? And use two equations?
What you found was a relation for diffraction minima, and nothing to do with zoom ratios. I'm not really sure what a "zoom ratio" is, are you talking about the ratio of magnification for the x-axis (along the mirror surface) to that along the y-axis? That would be relatively simple, it would just be the ratio in the two radii of curvature because that's related directly to the ratio of focal lengths and therefore to the ratio of magnifications. Aside from that, when you're talking about astigmatic mirrors you're getting deep into specialized optics. An actual optical person would be able to tell you more because they speak the language. Perhaps someone in astronomy.